Tuesday, May 17, 2011

Håstad to receive the 2011 Gödel Prize

Håstad is the fourth person to win the prize twice (after Shafi Goldwasser, Mario Szegedy and Sanjeev Arora). He won the 1994 Gödel Prize for his switching lemma and tight bounds on parity for low-depth circuits.

This is a landmark paper in computational complexity, specifically, the study of approximation properties of NP-hard problems. It improves on the PCP Theorem (recognized in a previous prize in 2001) to give novel probabilistic verifiers that can check membership proofs for NP languages while reading very few bits in them — as little as 3 bits. The existence of such verifiers implies that existing approximation algorithms for several problems such as MAX-3SAT cannot be improved if P is different from NP. In other words, there is a "threshold" approximation ratio which is possible to achieve in polynomial time, but improving upon which is NP-hard. Before this paper such "optimal" inapproximability results seemed beyond reach. The Fourier analytic techniques introduced in this paper have been adapted in dozens of other works, and are now taught in graduate courses in computational complexity. They also directly influenced subsequent work, such as the formulation of the unique games conjecture for proving further optimal inapproximability results, and lower bounds for geometric embeddings of metric spaces.

Hastad's work is great, but the citation should be clarified slightly: 3-query PCPs (with constant soundness gap) followed already from the original PCP theorem. What Hastad's work does is get essentially the optimal soundness gap for such PCPs, namely 1-vs.-(7/8 + eps).